\(\int \frac {1+x^4}{(-1+x^4) \sqrt {-1-x^2+x^4}} \, dx\) [162]

Optimal result

Integrand size = 27, antiderivative size = 19 \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {-1-x^2+x^4}} \, dx=-\arctan \left (\frac {x}{\sqrt {-1-x^2+x^4}}\right ) \]

[Out]

-arctan(x/(x^4-x^2-1)^(1/2))

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2137, 210} \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {-1-x^2+x^4}} \, dx=-\arctan \left (\frac {x}{\sqrt {x^4-x^2-1}}\right ) \]

[In]

Int[(1 + x^4)/((-1 + x^4)*Sqrt[-1 - x^2 + x^4]),x]

[Out]

-ArcTan[x/Sqrt[-1 - x^2 + x^4]]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 2137

Int[((u_)*((A_) + (B_.)*(x_)^4))/Sqrt[v_], x_Symbol] :> With[{a = Coeff[v, x, 0], b = Coeff[v, x, 2], c = Coef
f[v, x, 4], d = Coeff[1/u, x, 0], e = Coeff[1/u, x, 2], f = Coeff[1/u, x, 4]}, Dist[A, Subst[Int[1/(d - (b*d -
a*e)*x^2), x], x, x/Sqrt[v]], x] /; EqQ[a*B + A*c, 0] && EqQ[c*d - a*f, 0]] /; FreeQ[{A, B}, x] && PolyQ[v, x
^2, 2] && PolyQ[1/u, x^2, 2]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\frac {x}{\sqrt {-1-x^2+x^4}}\right ) \\ & = -\arctan \left (\frac {x}{\sqrt {-1-x^2+x^4}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {-1-x^2+x^4}} \, dx=-\arctan \left (\frac {x}{\sqrt {-1-x^2+x^4}}\right ) \]

[In]

Integrate[(1 + x^4)/((-1 + x^4)*Sqrt[-1 - x^2 + x^4]),x]

[Out]

-ArcTan[x/Sqrt[-1 - x^2 + x^4]]

Maple [A] (verified)

Time = 7.45 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

method	result	size
elliptic	\(\arctan \left (\frac {\sqrt {x^{4}-x^{2}-1}}{x}\right )\)	\(18\)
default	\(\frac {\arctan \left (\frac {-1+i+\left (1+i\right ) x^{2}+\left (-1-2 i\right ) x}{\sqrt {x^{4}-x^{2}-1}}\right )}{2}-\frac {\arctan \left (\frac {-1+i+\left (1+i\right ) x^{2}+\left (1+2 i\right ) x}{\sqrt {x^{4}-x^{2}-1}}\right )}{2}\)	\(66\)
pseudoelliptic	\(\frac {\arctan \left (\frac {-1+i+\left (1+i\right ) x^{2}+\left (-1-2 i\right ) x}{\sqrt {x^{4}-x^{2}-1}}\right )}{2}-\frac {\arctan \left (\frac {-1+i+\left (1+i\right ) x^{2}+\left (1+2 i\right ) x}{\sqrt {x^{4}-x^{2}-1}}\right )}{2}\)	\(66\)
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {x^{4}-x^{2}-1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (x -1\right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )}{2}\)	\(74\)

[In]

int((x^4+1)/(x^4-1)/(x^4-x^2-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

arctan(1/x*(x^4-x^2-1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.58 \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {-1-x^2+x^4}} \, dx=-\frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {x^{4} - x^{2} - 1} x}{x^{4} - 2 \, x^{2} - 1}\right ) \]

[In]

integrate((x^4+1)/(x^4-1)/(x^4-x^2-1)^(1/2),x, algorithm="fricas")

[Out]

-1/2*arctan(2*sqrt(x^4 - x^2 - 1)*x/(x^4 - 2*x^2 - 1))

Sympy [F]

\[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {-1-x^2+x^4}} \, dx=\int \frac {x^{4} + 1}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt {x^{4} - x^{2} - 1}}\, dx \]

[In]

integrate((x**4+1)/(x**4-1)/(x**4-x**2-1)**(1/2),x)

[Out]

Integral((x**4 + 1)/((x - 1)*(x + 1)*(x**2 + 1)*sqrt(x**4 - x**2 - 1)), x)

Maxima [F]

\[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {-1-x^2+x^4}} \, dx=\int { \frac {x^{4} + 1}{\sqrt {x^{4} - x^{2} - 1} {\left (x^{4} - 1\right )}} \,d x } \]

[In]

integrate((x^4+1)/(x^4-1)/(x^4-x^2-1)^(1/2),x, algorithm="maxima")

[Out]

integrate((x^4 + 1)/(sqrt(x^4 - x^2 - 1)*(x^4 - 1)), x)

Giac [F]

\[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {-1-x^2+x^4}} \, dx=\int { \frac {x^{4} + 1}{\sqrt {x^{4} - x^{2} - 1} {\left (x^{4} - 1\right )}} \,d x } \]

[In]

integrate((x^4+1)/(x^4-1)/(x^4-x^2-1)^(1/2),x, algorithm="giac")

[Out]

integrate((x^4 + 1)/(sqrt(x^4 - x^2 - 1)*(x^4 - 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1+x^4}{\left (-1+x^4\right ) \sqrt {-1-x^2+x^4}} \, dx=\int \frac {x^4+1}{\left (x^4-1\right )\,\sqrt {x^4-x^2-1}} \,d x \]

[In]

int((x^4 + 1)/((x^4 - 1)*(x^4 - x^2 - 1)^(1/2)),x)

[Out]

int((x^4 + 1)/((x^4 - 1)*(x^4 - x^2 - 1)^(1/2)), x)